How to use Rigid Body Physics after following path constraint?
Leave the Dynamic enabled, and keyframe Animated instead, on the same frames as you keyframe the Follow Path constraint. Keyframe it to on at the last frame where the constraint influence is 1.0 and to off on the frame after, where the influence is set to 0.0. Hover the mouse on the checkbox and press I to keyframe it.


I changed the Shape setting in the Rigid Body Collisions on the plane from Convex Hull to Mesh, but that isn't strictly necessary if you're ok with the buckler bouncing a little above the plane.
  

To bake the physics, go to the Scene window and click Bake or Bake All Dynamics in the Rigid Body Cache panel. And if you're only going to render 120 frames, as set in the blend file, you may as well change the end frame for the physics too. No use baking frames you're not going to render anyway.


Here's a short animated gif. I apologise for the quality. I used Blender's built-in screencast, and apparently it wasn't able to keep up with the speed at which the buckler was moving.